This phase I/II trial studies the side effects and best dose of sirolimus when given together with cisplatin and gemcitabine hydrochloride and to see how well they work in treating patients with bladder cancer. Biological therapies, such as sirolimus, may stimulate or suppress the immune system in different ways and stop tumor cells from growing. Drugs used in chemotherapy, such as cisplatin and gemcitabine hydrochloride, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Giving sirolimus together with cisplatin and gemcitabine hydrochloride may be an effective treatment for bladder cancer.